Establishment of Model of Damping Mechanism for the Hard-coating Cantilever Plate

Transcription

1 Etalihment of Model of Damping Mehanim for the Hard-oating Cantilever Plate Rong Liu 1, Ran Li 1, Wei Sun 1* 1 Shool of Mehanial Engineering & Automation, Northeatern Univerity, Shenyang , China Atrat: Contrained y the aility of teting damping, there i no effetive method to eparate the damping ontriution of hard oating from the ytem damping of ompoite truture, whih make the etalihment of model of damping mehanim eome very diffiult. In thi paper, aed on eparating the damping ontriution of hard oating, the method of reating the damping mehanim model of antilever ompoite plate wa tudied. Firt of all, the antilever plate efore and after oating were teted and the dynami harateriti parameter, uh a natural frequeny, damping ratio, viration repone and o on, were otained. Moreover, from the analyi of the torage and diipation energy in the unoated and oated plate, the damping ontriution of hard oating in the whole ytem damping wa onfirmed. Finally, aed on the Lagrange equation, the dynami model of antilever ompoite plate wa reated onidering oth the material damping of hard oating and the remaining equivalent viou damping of ytem. The orretne of model wa verified y experiment reult. The propoed method an e further applied to more omplex truture and provide the referene to the tudy of viration redution mehanim and energy diipation mehanim of hard oating. Keyword: Hard oating; Cantilever thin plate; Damping mehanim; Analyi model; Baement exitation 1 Introdution Hard oating an e ued to redue the viration repone of thin-walled truture in high temperature, high orroion environment, and o road attention ha een arien in reent year [1-3]. In order to etter implement hard oating damping viration, the damping mehanim of hard oating need to e aquired. Here, the damping mehanim i that uing the model to explain the reaon of viration redution of hard oating. On the ai of otaining the damping mehanim, the optimization deign of viration redution ued hard oating an e exeuted. Now, mot of the exiting tudie [4-8] on damping mehanim of hard oating ome from miro material iene. For example, Taini [4] reated a phenomenologial model ued to haraterize the elati propertie of hard oating material, and the model an reprodue the ai feature of the oerved * Correponding author: Wei Sun (weiun@mail.neu.edu.n) 14 ISSN (Print), (Online)

2 International Journal of Smart Engineering, Volume 1, Iue, 017 damping ehavior of ompoite truture. To undertand the peifi mehanim of hard oating, Torvik [5] developed a lip damping model whih an provide a atifatory analytial repreentation of the damping and tiffne of hard-oating material. Al-Ru [6] propoed a miromehanial theoretial and omputational model to ae the main miromehanial mehanim reponile for the experimentally oerved phenomenon. From the aove reearhe, mot holar think the internal frition among miro partile of hard oating i the aue of viration redution of hard-oating ompoite truture. However, it i not enough to reearh the damping mehanim of hard oating ompletely aed on miro material iene. It i neeary to reate the dynami model of the hard oating ompoite truture, and then reearh the damping mehanim of hard oating from the angle of viration iene. Some holar have attempted to determine the viration redution mehanim of hard oating y reating dynami model of ompoite truture. For example, Patia [9] analyzed the dynami propertie of antilever eam ytem partially oated with hard oating uing the eam model, and predited the nature harateriti and the damping parameter of the ompoite eam. Yen [10] developed an analytial proedure to tudy the dynami ehavior of eam oated with hard oating. But the tatu of urrent reearh i till unale to meet the need of damping deign of hard oating. To reate the analyi model whih an aurately repreent dynami harateriti of hard oating ompoite truture, a lot of experimental reult are needed, uh a nature frequeny, damping and viration repone for the unoated and oated truture. However, ompared with teting nature frequeny and viration repone, there i no aurate method ued to tet damping [11]. The general method of identifying damping are the half power andwidth [1] or free viration attenuation method [13], whih are all aed on the aumption of equivalent viou damping. The damping otained y the two meaure method i an integrated value, uually named a damping ratio, whih inlude the material damping of hard oating, oundary ondition damping and fluid damping in air, et. The damping ontriution of hard oating i not effetively eparated from the integrated value. Therefore, it i very diffiult to reate the damping mehanim model of hard-oating ompoite truture from the angle of viration iene. In thi reearh, the antilever thin plate oated Mg-Al hard oating under the aement exitation wa taken a tudy ojet, and the damping mehanim model of hard-oating ompoite truture wa reated aed on otaining the damping ontriution of hard oating from the ytem damping of ompoite plate. In etion, the antilever plate efore and after oating were teted and the dynami harateriti parameter, uh a natural frequeny, damping ratio, viration repone and o on, were otained. In etion 3, from the analyi of the torage and diipation energy in the unoated and oated plate, the damping ontriution of hard oating wa eparated. In etion 4, the method of otain the damping ontriution of hard oating wa propoed. Finally, in etion5, aed on the Lagrange equation, the dynami model of antilever ompoite plate wa reated onidering oth the material damping of hard oating and the remaining equivalent viou damping of ytem, and the orretne of model wa alo verified y experiment. ISSN (Print), (Online) 15

3 International Journal of Smart Engineering, Volume 1, Iue, 017 The experiment of antilever thin plate efore and after oating Mg-Al hard oating i a peudo alloy oating, and it i uually ued a anti-frition and anti-eroion oating. Thi oating an alo e ued a damping oating eaue there are many rak and void in the oating (hown in Fig.1) and the internal frition among oating partile will produe damping effet. In thi tudy, the titanium plate oated with Mg-Al hard oating in one ide (hown in Fig.) i hoen a experimental ojet. Fig.1. Surfae morphology of Mg-Al hard oating (a) unoated plate ()oated plate Fig.. The experiment ojet The geometrial and material parameter of the plate are lited in Tale.1, and the experiment ytem i hown in Fig.3. Thee parameter in Tale.1 are otained y the atual tet or alulation. For example, viration eam method [14] wa adopted to identify the Young' modulu of Mg-Al oating. The plate i fixed on the fixture and lamping area i 0mm. The purpoe of experiment i to otain the natural frequeny, damping ratio and viration repone of the antilever plate efore and after oating. The tet equipment inlude LMS SCADAS aquiition front-end, King Deign EM-1000F haker and B&K 4517 light aelerometer, et. It hould e noted that it i true that the ma of aelerometer will impat the dynami of ytem, ut in thi work, the ma impat i ignored. Beaue the ma of B&K 4517 aelerometer i only 0.6g, in addition, the aelerometer i plaed on the lower part of antilever plate, o the impat of aelerometer ma on the whole dynami ytem i mall and an e omitted. 16 ISSN (Print), (Online)

4 International Journal of Smart Engineering, Volume 1, Iue, 017 Viration haker Hard oating titanium plate Diretion of exitation Aelerometer Fig.3. The oated thin plate under the ae exitation Tale 1. Geometrial and material parameter of titanium plate and hard oating Type of material Sutrate Ti-6Al-4V Hard oating Mg-Al Length (mm) Width (mm) Thikne (mm) Young' modulu (GPa) Denity (kg/m3) Poion' ratio Amplitude (m/s ) Time (S) Frequeny (Hz) Fig.4. The 3D waterfall figure otained y weep exitation for the 6-order of oated plate ISSN (Print), (Online) 17

5 International Journal of Smart Engineering, Volume 1, Iue, 017 Tale. The natural frequenie of unoated and oated plate/hz Order Statu unoated(a 1) oated(b 1) differene (B 1-A 1/A 1) -.% -1.8% -4.3% -5.3% -1.1% -1.1% Tale 3. The modal damping ratio of unoated and oated plate/% Order Statu unoated(a ) oated(b ) differene (B -A /A ) 54.1% 6.3% 9.1% 48.3% 66.7% 7.7% Tale 4. The viration repone of unoated and oated plate/m/ Order Statu unoated(a 3) oated(b 3) differene (B 3-A 3/A 3) -8.% -7.5% -6.8% -34.7% -51.% -31.8% The natural frequenie of antilever plate are onfirmed from 3D waterfall figure otained y weep exitation, an example aout 3D waterfall figure an e een in Fig.4, and the firt 6 order natural frequenie of unoated and oated plate are lited in Tale. The method of half power andwidth i adopted to otain the modal damping ratio and the relevant teting reult are lited in Tale 3. The plate i exited y eah order reonane frequeny, exitation amplitude i 1g, when ytem ome to e teady-tate, the reonant repone of antilever plate ould e otained and the reult are lited in Tale 4. A an e een from the tet reult, ompared with unoated plate, the natural frequenie of the plate oated with Mg-Al hard oating hange lightly,the igget differene i le than 6%. For the firt 6 order, eah order damping ratio of the oated plate inreae and the reonant repone are redued. Among them, the 1, 4, 5, 6 order, hanged ignifiantly. It indiate that the hard oating poee etter effet of viration redution. 3 The analyi of torage and diipation energy of thin plate efore and after oating 18 ISSN (Print), (Online)

6 International Journal of Smart Engineering, Volume 1, Iue, 017 For the antilever plate under ae exitation hown in Fig.3, tored energie are primarily kineti energy and potential energy, and diipation energie inlude the fritional diipation of lamping area of the plate, energy diipation of material damping, and energy diipation from air damping with movement of the plate. Under the ame exitation amplitude, there are ignifiantly different of the torage and diipation energy etween the unoated and oated plate. Set U a torage energy of eah yle of antilever plate (utrate ytem), it an e expreed a: U Uˆ ˆ T (1) U ˆ i maximum train energy, T ˆ i maximum kineti energy. Alo, et D a diipation energy of eah yle of antilever plate, it an e expreed a D Da Dm Dn () D i diipation energy of lamping area of the antilever plate, i a diipation energy of material damping of the plate, damping. Thu, the lo fator of unoated ytem an e repreented a: D D D D n Dm D i diipation energy from air a m n (3) U U If the antilever plate ytem i in the reonane tate under a ertain natural frequeny exitation, the lo fator of ytem i the modal lo fator, and modal lo fator i twie of the modal damping ratio, it an e written a: (4) i the modal damping ratio of the unoated plate, whih i orreponded with the damping tet value lited in Tale.3. After oating, the torage and diipation energy will e hange. Set U a torage energy of eah yle of oated plate and it an e expreed a: U U U (5) U i torage energy of the utrate, Similar with the analyi of unoated plate, plate and an e expreed a: a m n U i torage energy of the hard oating. D i et a diipation energy of oated D D D D D (6) D a, D m, D n are the diipated energie of lamping area, the utrate material and air damping repetively and D i diipation energy of hard oating material. Similarly, the lo fator of the oated ytem an e written a: D D D D D U ( U U ) a m n (7) ISSN (Print), (Online) 19

7 International Journal of Smart Engineering, Volume 1, Iue, 017 In the reonane tate, the lo fator i modal lo fator. The relationhip etween the modal lo fator and modal damping ratio i: (8) i the modal damping ratio of oated plate, whih i alo orreponded with the damping tet value lited in Tale.3. Eq.(7) an e further expreed a: D / U ( D a D m D n ) / U ( U / U 1) ( U / U 1) If et It an yield: ( D D D ) D D D U U a m n a m n U / U 1 U / U 1 U / U 1 U / U 1 derie the ontriution of hard oating in the whole lo fator of ytem (or the whole ytem damping). Here, i named a the remaining equivalent viou damping of ytem, eaue it exlude the effet of the material damping of hard oating. It an e een that the damping ontriution of hard oating ha een eparated from the whole ytem damping. In the uequent analyi model, it an e aumed that there only inlude the material damping of hard oating (Generally, the material damping of utrate i muh le than hard oating.) and the remaining equivalent viou damping. Referring to the Eq.(10), it an e thought, that i, the ytem damping of unoated plate an e ued in the analyi model of oated plate. It i worth noting that the meaning of lo fator ome from maro viration iene, ut it i almot onitent with the definition of material iene (hown in Eq.(19) of etion 5), only, for the different order of thin plate, will e aigned different value. A for, it i different with the definition of material iene, eaue it inlude not only material damping of utrate ut alo other damping, uh a fritional damping in the lamping area, air damping, et, o it i named a the remaining equivalent viou damping in thi work. (9) (10) (11) 4 Otain the damping ontriution of hard oating in the whole ytem damping By defining the torage energy ratio R U / U etween hard oating and utrate, Eq.(11) an e expreed a: ( R 1) (1) R 0 ISSN (Print), (Online)

8 International Journal of Smart Engineering, Volume 1, Iue, 017 Thu, a long a the torage energy ratio R i known, the damping ontriution of hard oating in the whole ytem damping an e otained. Aording to Eq.(1), the torage energy ratio R an alo e expreed a the maximum train energy ratio or maximum kineti energy ratio, and i hown a follow: Uˆ ˆ T R (13) U ˆ T ˆ Uˆ and repetively, and Uˆ are the maximum train energy of the hard oating and utrate T ˆ and T ˆ are the maximum kineti energy of the hard oating and utrate repetively. Beaue energy an e uperimpoed, the total train energy and kineti energy of the hard oating ompoite truture are: Uˆ Uˆ ˆ U (14) Tˆ Tˆ T ˆ (15) Aording to the Rayleigh quotient, the quare of natural frequeny of truture ytem i equal to the ratio of the maximum potential energy (Here, the train energy i equivalent to potential energy) and the maximum kineti energy, thu it an yield: f Uˆ Tˆ ˆ ˆ 1U U (16) f Uˆ Tˆ 1Tˆ Tˆ f and f are the natural frequenie of unoated and oated plate. In Eq. (16), the ratio of kineti energy an alo e expreed a the ratio of ma, that i: T ˆ T ˆ H / H (17) and H and H are the thikne of hard oating and utrate repetively, and are the denity of hard oating and utrate. So, Eq. (16) an e expreed a: Uˆ f H R (1 ) 1 Uˆ f H Eq.(18) how that if getting the natural frequeny of unoated and oated plate, the thikne and denity of hard oating and utrate, the torage energy ratio an e otained. Furthermore, the damping ontriution of hard oating orreponding eah order lo fator of ytem an e otained y utituting the Eq.(18) into Eq.(1). (18) 5 Creating the damping mehanim model of antilever ompoite plate 5.1 Analyi of hard-oating ompoite plate under ae exitation A antilever thin plate with hard oating on one ide i hown in Fig.5(a). It length i a and the width i. The thin plate i ujeted to the ae exitation denoted y u () t g at it lamping end. Fig. 5() how it ro etion and et xy oordinate plane loate in ISSN (Print), (Online) 1

9 International Journal of Smart Engineering, Volume 1, Iue, 017 the neutral urfae. i the ditane etween the interfae onding of oating-utrate and the neutral urfae. z y u () t g z a Metal utrate Hard oating (a) Hard-oated ompoite plate x H H Hard oating Metal utrate () Cro etion of ompoite plate Fig. 5. Hard-oated ompoite plate under ae exitation The elati modulu of hard oating i expreed a omplex modulu and hown a: * E E (1 i ) (19) * * refer to omplex value, E are the omplex modulu of the hard oating, and E, are the orreponding Young modulu (or torage modulu) and the lo fator repetively. For the normal material, the i uually ontant, ut here H H x i utituted aording to every order, that i, inputting the identified value hown in Eq.(1). Thi reflet the material damping of hard oating on the ontriution of eah order modal of ompoite plate. Then, the omplex hear modulu of hard oating an e expreed a: * * E G (0) 1 * G i the omplex hear modulu of hard oating, and i orreponding Poion' ratio. The ditane etween the interfae and neutral urfae an e determined a [15] : EH EH (1) E H E H E i Young modulu of metal utrate. It i aumed that the ae exitation at the lamping end of the plate i harmoni and expreed a: i u () t U e t () g U G i the amplitude of ae exitation and i the angular frequeny. Hene, the total diplaement of an aritrary point on the antilever plate an e determined a: ( x, y, t)= u ( t) + w( x, y, t) (3) the w( x, y, t ) refer to the defletion of any point. If the hard-oated ompoite plate atifie the ai aumption of the thin plate theory, thu the train energy of the plate i G g ISSN (Print), (Online)

10 International Journal of Smart Engineering, Volume 1, Iue, 017 Uˆ 1 x x y y xy xy V 1 w w w w w L1 L L 3 da xy A x y x y V,A i the volume and area of the plate and L1, L, L 3 are E N E N * L G, L E N 3 H H N1 H 1 dv E N * L 4G N 4G N * N H H H 3 are hear modulu and Poion' ratio of metal utrate repetively. Eq.(4) and Eq.(5) how that the energy diipation due to the material damping of hard oating ha een inluded in the alulation of train energy of ytem. The kineti energy of ompoite plate i: 1 w t A (4) (5) Tˆ H H da (6) Aording to the two-dimenional eam funtion method, the defletion of the hard-oated ompoite plate an e approximately expreed a: R S,, w x y t X x Y y a t (7) r1 1 r r Xr x refer to the r-order modal hape funtion of lamped-free eam, Y y refer to the -order modal hape funtion of free -free eam, ar t i the ontriution oeffiient of eah eam funtion, and R, S are the upper limit of mode intereted. Set L T ˆ ˆ U,applying Lagrange equation yield: d L L D p1,,3,, R 0 (8) dt a pk t a pk t a pk t k 1,,3,, S D i the diipation funtion of the ompoite plate whih ha exluded the ontriution of hard-oating material, it ould alo e een a diipation energy of the unoated plate, that i: R S 1 D D j ( X r xy yar ( t)) r1 1 i the damping oeffiient of the j-order, it an e expreed a: j j, jj / m (9),, are the remaining equivalent viou damping and natural frequeny of j j the j-order of unoated plate repetively, m i the ma of per unit area. Beaue the plate i exited y a ingle frequeny, it an e aumed that i t a t a e (30) r Then, the defletion of the hard-oating ompoite plate an e re-written a: r ISSN (Print), (Online) 3

11 International Journal of Smart Engineering, Volume 1, Iue, 017 R S r r t (31) r1 1 i,, w x y t X x Y y a e Aording to Eq. (8), a et of linear algerai equation an e otained, in matrix form: K ic M ic a q (3) 1 a a a a a a a (33) 11 1S 1 S r RS repreent the repone of ytem, the oeffiient matrix K i the tiffne matrix, M i the ma matrix, C 1 i the material damping matrix of hard oating, C i the remaining equivalent viou damping of ytem,q i the vetor of the exitation fore. From the Eq.(3), the repone vetor a an e otained orreponding to a peifi exiting frequeny. Sutituting a to the Eq.(7), the defletion of any point at the hard-oated ompoite plate an e aquired. Finally, the viration repone under ae exitation i ahieved through Eq. (3). T 5. Solution of viration repone Correponding to the experiment hown in etion, the reated analyi model i ued to olve the viration repone. By omparing with the experimental reult, the orretne of model an e verified. The eah order torage energy ratio etween hard oating and utrate an e otained y utituting the natural frequenie of unoated and oated plate into the Eq.(18). Moreover, utituting torage energy ratio into Eq.(1), the damping ontriution of hard oating in eah order ytem damping an e otained. All the reult are lited in Tale.5. Tale 5. The firt 6 order torage energy ratio and the damping ontriution of hard oating in eah order ytem damping Order torage energy ratio R Modal lo fator /% Then, utituting the firt 6 order modal damping ratio of unoated plate lited in Tale.3 and the firt 6 order lo fator of hard oating lited in Tale.5 into the analyi model, the viration repone of oated plate are otained. Here, exitation amplitude i till 1g in the analyi model and the viration piking point i onitent with experiment, the oordinate i : x=0.03m,y=0.08m. The otained analyi reult are lited in Tale.6. It an e een that there are only a mall amount of differene etween the experimental reult and analyi reult otained y the reated model and the maximum differene i le than 6.5%(orreponding to the 5th order), o the orretne of the reated model i verified. In addition, it hould e explained that the eond order and the fourth order reonant repone an not e olved y the reated model, eaue the model an not effetively imulate the torion exitation. The firt 6 4 ISSN (Print), (Online)

12 International Journal of Smart Engineering, Volume 1, Iue, 017 order modal hape of oated antilever plate are hown in Fig.6, whih are otained y finite element method ued ANSYS oftware. It an e een that the nd and 4th modal hape are torion viration. Tale 6. Reonant repone omparion etween experiment and analyi for the oated antilever plate/ m/ Order Statu experiment(a 4) analyi(b 4) A B / A 5.8%.83% 6.33% 0.158% (a) the 1t order () the nd order () the 3rd order (d) the 4th order (e) the 5th order (f) the 6th order Fig.6. The firt 6 order modal hape of oated antilever plate 6 Conluion Baed on eparating the damping ontriution of hard oating from the ytem damping otained y tet, the method of reating the damping mehanim model of antilever ompoite plate wa tudied and reahed the following onluion: (1) The antilever thin plate oated with Mg-Al hard oating on one ide wa teted, the tet reult how eah order damping ratio inreae and viration repone are redued and ome order hanged ignifiantly. It indiate that the hard oating poee etter effet of viration redution. ISSN (Print), (Online) 5

13 International Journal of Smart Engineering, Volume 1, Iue, 017 () By analyzing the torage and diipation energy of ompoite truture and utilizing the definition of lo fator, the damping ontriution of hard oating an e eparated from the whole ytem damping and the relevant formula wa derived. (3) The damping of oated ompoite plate ytem inlude the material damping of the hard oating and other damping that an e named a remaining equivalent viou damping. In addition, the remaining equivalent viou damping an e replaed y the ytem damping of unoated plate. The pratie indiate that it ould not aue a large analyi error. (4) Utilizing the material damping of hard oating and the remaining equivalent viou damping, the analytial model of hard-oating ompoite plate an e reated, whih an effetively imulate the dynami of the real truture. The alulation reult of the firt 6 order reonant repone (exept for the and 4 order) demontrate that the maximum error i le than 6.5%. For the reated analyi model in thi tudy, the damping ontriution of hard oating wa eparated from the ytem damping of ompoite plate, and then the goal of reating the model of damping mehanim from viration iene wa ahieved. Furthermore, y analyzing the impat of thikne, Young modulu and lo fator of hard oating on the effet of viration redution, the reated model alo an provide the referene for hooing and fariating the hard-oating material, o parameter analyi hould e done in next work. Aknowledgment Thi projet i upported y National Natural Siene Foundation of China (Grant No ). Referene 1. C Blakwell, A Palazotto, TJ George, et al, The evaluation of the damping harateriti of a hard oating on titanium, Shok and Viration 14 (1) (007) F Ivani, A Palazotto, Experimental onideration for determining the damping oeffiient of hard oating, Journal of Aeropae Engineering 18 (005) GY Du, DC Ba, Z Tan, et al, Viration damping performane of ZrTiN oating depoited y ar ion plating on TC4 Titanium alloy, Surfae and Coating Tehnology 9 (013) N Taini, S Patia, K Lamrinou, Cerami oating: a phenomenologial modeling for damping ehavior related to mirotrutural feature, Material Siene and Engineering: A 44 (1) (006) PJ Torvik, A Slip Damping Model for Plama Sprayed Cerami, Journal of Applied Mehani 76 (6) (009) RKA Al-Ru,AN Palazotto, Miromehanial theoretial and omputational modeling of energy diipation due to nonlinear viration of hard erami oating with mirotrutural reurive fault, International Journal of Solid and Struture 47 (010) L Yu, Y Ma, C Zhou, et al, Damping effiieny of the oating truture. International Journal of Solid and Struture 4 (005) ISSN (Print), (Online)

14 International Journal of Smart Engineering, Volume 1, Iue, L Yu, Y Ma, C Zhou, et al, Damping apaity and dynami mehanial harateriti of the plama-prayed oating, Material Siene and Engineering: A 407(005) S Patia, R William, Hard damping oating: material propertie and FE predition method,proeeding of 8 th National Turine Engine High Cyle Fatigue Conferene, (003). 10. HY Yen, MHH Shen, Paive viration uppreion of eam and lade uing magnetomehanial oating, Journal of ound and viration 45 (001) AS Phani, J Woodhoue, Viou damping identifiation in linear viration, Journal of Sound and Viration 303 (007) PJ Torvik, On etimating ytem damping from frequeny repone andwidth, Journal of Sound and Viration 330 (011) F Magalhãe, Á Cunha, E Caetano, et al, Damping etimation uing free deay and amient viration tet, Mehanial Sytem and Signal Proeing 4 (010) ASTM International, ASTME tandard tet method for meauring viration-damping propertie of material, (005). 15. G Parthaarathy, CVR Reddy, N Ganean, Partial overage of retangular plate y unontrained layer damping treatment, Journal of Sound and Viration 10 (1985) ISSN (Print), (Online) 7